The Batchelor Spectrum of Passive Scalar Turbulence in Stochastic Fluid Mechanics at Fixed Reynolds Number

In 1959 Batchelor predicted that the stationary statistics of passive scalars advected in fluids with small diffusivity k should display a power spectrum along an inertial range contained viscous-convective fluid model. This prediction has been extensively tested, both experimentally and numerically, is core scalar turbulence. this article we provide rigorous proof version Batchelor's limit when subjected to spatially smooth, white-in-time stochastic source by 2D Navier-Stokes equations or 3D hyperviscous forced sufficiently regular, nondegenerate forcing. Although our results hold for at arbitrary Reynolds number, value fixed throughout. Our rely on quantitative understanding Lagrangian chaos mixing established recent works. Additionally, limit, obtain statistically stationary, weak solutions stochastically advection problem without diffusivity. These are almost-surely not locally integrable distributions nonvanishing average anomalous flux satisfy all scales. We also prove Onsager-type criticality result shows no such dissipative, little more regularity can exist. © 2021 Wiley Periodicals LLC.